Moment Generating Function of the Product of Two Independent Standard Normal Random Variables

In probability theory, the moment generating function (MGF) is a powerful tool that provides a way to compute moments of a random variable. This article delves into the computation of the moment generating function for the product of two independent standard normal random variables. We will walk through the derivation step-by-step and highlight the key properties of this interesting distribution.

Introduction to Standard Normal Distribution

Let's start by understanding two independent standard normal random variables (X) and (Y). A random variable is called standard normal if its probability density function (PDF) is given by:

$$f_X(x) frac{1}{sqrt{2pi}} e^{-frac{x^2}{2}} quad text{and} quad f_Y(y) frac{1}{sqrt{2pi}} e^{-frac{y^2}{2}} quad text{for} quad x, y in mathbb{R}$$

Since (X) and (Y) are independent, their joint probability density function (JPDF) is simply the product of their individual PDFs:

$$f_{X,Y}(x, y) f_X(x) cdot f_Y(y) frac{1}{2pi} e^{-frac{x^2 y^2}{2}} quad text{for} quad x, y in mathbb{R}$$

Computing the Moment Generating Function

The moment generating function (MGF) of a random variable (Z) is defined as:

$$m_Z(t) E[e^{tZ}] int_{-infty}^{infty} int_{-infty}^{infty} e^{txy} f_{X,Y}(x, y) dx dy$$

Substituting the joint PDF into the definition of the MGF, we get:

$$m_{X,Y}(t) int_{-infty}^{infty} int_{-infty}^{infty} e^{txy} cdot frac{1}{2pi} e^{-frac{x^2 y^2}{2}} dx dy$$

To simplify the evaluation of this double integral, we use a change of variables and complete the square. Let's first rewrite the exponent:

$$m_{X,Y}(t) frac{1}{2pi} int_{-infty}^{infty} int_{-infty}^{infty} e^{-x^2 - 2txy - frac{y^2}{2}} dx dy frac{1}{2pi} int_{-infty}^{infty} int_{-infty}^{infty} e^{-x - ty} e^{-frac{y^2 (1 - t^2)}{2}} dx dy$$

By making a change of variables:

$$u frac{x - ty}{sqrt{2}} quad text{and} quad v y sqrt{frac{1 - t^2}{2}}$$

The Jacobian of this transformation is:

$$frac{partial (u, v)}{partial (x, y)} frac{1}{sqrt{1 - t^2}}$$

So the change of variables yields:

$$m_{X,Y}(t) frac{1}{2pi} cdot frac{2}{sqrt{1 - t^2}} int_{-infty}^{infty} int_{-infty}^{infty} e^{-u^2} e^{-v^2} du dv$$

This is now a product of two Gaussian integrals:

$$m_{X,Y}(t) frac{1}{pi sqrt{1 - t^2}} left( int_{-infty}^{infty} e^{-u^2} du right)^2 frac{1}{pi sqrt{1 - t^2}} cdot (sqrt{pi})^2 boxed{frac{1}{sqrt{1 - t^2}}}$$

Conclusion

In conclusion, the moment generating function for the product of two independent standard normal random variables is a useful representation that simplifies the computation of moments. This result has several interesting applications in statistics and probability theory.